Spatial Accuracy Assessment of Digital Surface Models: A Probabilistic...
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Spatial Accuracy Assessment of Digital Surface Models:A Probabilistic Approach
André JalobeanuCentro de Geofisica de Evora,
Portugal
part of the SPACEFUSION Project (ANR, France)
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Outline
Goals and objectives
Uncertainty prediction
Spatial variability
Why we need uncertainties
Bayesian inference
Forward model
2-step hierarchical inversion
Results from real data
Future work
!
f
Y
"
#, $%
&
h
resolution
cameraposeparameters
internalcameraparameters
globalPSF
sensor sampling grid
noise
observedimage
geometricmapping
renderingcoefficients
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Goals and objectives
! Digital Surface Models with error maps
! Provide the uncertainties to allow for error propagation
! Automatic parameter estimation (e.g. orientation...)
! Dense disparity maps with sub-pixel accuracy
! Why use stereo optical images
! Availability, coverage, redundancy, price
! Requirements
! Raw and well-sampled images, (coarse) reference DSM
! Necessary tools
! Probability theory, signal processing, computer vision,applied math, and some Physics!
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Uncertainty prediction
! Classical approaches in photogrammetry
! Error assessment using reference data sets
! Global accuracy measures in general (RMS...)
! Some predictors exist (slope, land cover), but rely on validation
! Proposed method
! Predict, rather than assess the spatial accuracy
! Use all the available information (images, coarse DSMs)
! No need for a more accurate, reference DSM!(except when we need to validate the method)
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Why uncertainty is spatially variable
! Image quality
! Spatially variable noise, bad data
! Missing data
! Localized clouds and occlusions
! Scene information content
! Textureless areas
! Strong variations due to reflectance
adaptive uncertainty mapscarry important information! d
iffe
rence
image 1
image 2
because stereo cues are spatially variable!
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Why we need to evaluate uncertainties correctly
Avera
ge
Probabilistic fusion
Result #1
Result #2
! Data fusion
! Probabilistic framework to combine multiple uncertain objects
! Use variances, and also correlations between variables
! Error propagation
! From the observation noise to the end result!
! Strong dependence on the spatial variability of error:
•Flood hazard maps - avoid under- or overconfident estimates
•Fuzzy contour maps - display the spatial accuracy
probabilistic fusionvs. averaging
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Example: flood maps
disparity map [-10,10]
error map (std dev) [0,1]
">T P(">T)uniform error
P(">T)spatially variable
error map
different results!
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Yes we can compute uncertainty rigorously...thanks to Bayesian inference
p(! | observations) =p(observations | !)! p(!)
p(observations)
evidence(useful for model comparison)
likelihoodimage formation model
prior model(a priori knowledge
about the observed object)
parameters of interest(unknown solution)
OBJECTIVE:posterior probabilitydensity function (pdf)
•Eliminate the unwanted parameters (integration)
•Compute the optimal parameters of interest (optimization)
•Compute the related uncertainties (derivatives)
•Model selection and assessment (comparison)
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Basic ingredients & mathematical tools
! Forward modeling:
•All parameters are random variables
•Data - image formation model (rendering + degradations)
•Prior - object modeling
•Graphical models convenient for design and understanding
! Proposed Bayesian inference scheme:
•Marginalization or integration w.r.t. nuisance parameters
•Approximations (otherwise intractable)
•Deterministic optimization (for speed)
•Uncertainty evaluation
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Forward model1. Underlying 2D “reflectance map”
! Common reflected radiance map
! Model this map as a 2D image:
! Use the Nyquist-Shannon sampling theorem
•Frequency cut-off (limited optical resolution)
•Band-limited image representation using B-Splines
•Resampling through B-Spline interpolation
X Y1Y2
Spline kernelRepresentation - sum of splines
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Forward model2. Radiometric change model
! Spatially adaptive parametric model
! Multiplicative changes - include reflectance effects (non-Lambert, lighting variations), shadows, atmospheric attenuation, instrumental artifacts...
! Additive changes - include atmospheric haze, clouds, instrumental biases...
! Additive noise - approx. Gaussian, independent pixels
Y1 Y2
changes are spatially variable,so should be the the model parameters
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! Arbitrary disparity: surface deformation
! Applications: earthquakes, change detection...
! Constrained disparity for DSM reconstruction
! Epipolar lines if the exterior orientation is known!
! Rigorous sensor model for along-track stereo:
•"x very smooth
•"y directly related to the surface topography
example: planetary surface modeling
Self-similar process based on image gradients
Markov Random Field:spatial interactions btw. neighbors
Forward model3. Smoothness priors for disparities or surfaces
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Graphical model (forward model)
! Oriented graphical model or Bayesian network
! Easily build the joint probability density function (pdf)
! Use the graph structure for an efficient, hierarchical inversion
Y2
X!y
"
observed
images
disparity
maps
underlying
scene
CA,B
Y1
change
maps
smoothness
deterministic stochastic
DSM #
Ref
DSM!x
control
DSM
sensor
orientation
$ref
noise
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Inference: invert the forward model
! Marginalize all the nuisance variables (not of interest)
! Use explicit values for known parameters
! Estimate unknown parameters automatically and plug in the results
Y2
X!y
"
observed
images
disparity
maps
underlying
scene
CA,B
Y1
change
maps
smoothness
deterministic stochastic
DSM #
Ref
DSM!x
control
DSM
sensor
orientation
$ref
noise
P(DSM | data)
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Step 1 - disparity inference
Y2
X!y
"
observed
images
disparity
maps
underlying
scene
CA,B
Y1
change
maps
smoothness
deterministic stochastic
#
!x
coarse
deformation
P("y | Y1,Y2)
! Marginalization of A,B,C,Xapprox. by area-based matching
! Multigrid approachfor both disparity and image data(coarse to fine)
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How the disparity inference works
Iterative optimization of an energy functional
(nonlinear search: conjugate gradient, LBP, graph cuts...)
log P("x,"y | Y1,Y2) = D("x,"y,Y1,Y2) + Prior("x)+Prior("y)
data term smoothness penalty
self
vertical
horiz.
optimaldx,dy
optimal disparity + uncertainties
! Compute the posterior marginal pdfs
! Gaussian approximation: optimum & variance
(marginal pdf of 2 neighbors ! covariance)
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Step 2 - orientation via surface matching
P(# | "y, DSMref)
!y
disparity
map
DSM "
Ref
DSM
control
DSM
sensor
orientation
#ref
noise
#
DSMrefDSM
"y
Bayesiansurface matching
(cartesian coordinates!)
optimal orientation parameters
+ uncertainties (covariance matrix)
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Step 3 - probabilistic DSM and error structure
P(DSM | "y, #)
#
DSM
"y
inverse sensor model
(x,y,"y)
#
(X,Y,Z)
! Vertical errors:mostly due to errors on "y (transformed using #)Markov Random Field structure
! Planimetric errors: due to errors on # (transformed using x,y)highly correlated
From image spaceto cartesian ground frame
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Results - disparity inference
Area of interest: Nanedi Valles on Mars (location 6.0°N, 312°E)
Data used: Mars Express HRSC orbit 0905, stereo channels s1 and s2 (30 m ground resolution),
2048x4096 region of interest, raw unprojected data.
disparity map error map (std dev)[0=black,1=white]
input stereo images
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Results - surface matching, DEM generation
Local Linear Reconstruction
MOLA DEM(local ENU frame)
unit:meter
Reconstructed DEM
(local ENU frame)
N
Reference for local ENU frame: 6.0°N, 312°E, 3396200 m
Reference DEM (MOLA)
(local ENU frame)
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Results: DEM and error map
Reconstructed DEM (unit: meter)
local ENU frame
Error map (std dev) (unit: meter)
local ENU frame
Horizontal cross-section (Y=0)+ error envelopes (1 std dev)
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Results - disparity inference
disparity map [-20,20] error map (std dev)[0=black,1=white]
input stereo images(enhanced)
Data used: ALOS PRISM free sample data, stereo channels nadir and backward,
downsampled to 20 m ground resolution,
512x512 pixels region of interest, raw unprojected data
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Results - vector disparity mapRAW SPOT 5, multidate, 128x128 pixels @ 3.5m
change map correlation map change map
images Y1, Y2 [0,255]
x and y std dev maps
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Results - vector disparity mapRAW SPOT 5, multidate, 1024x1024 pixels @ 3.5m
Color-coded vector disparity map, linear correction applied
(area near Bam, Iran;across-track stereo pair)
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Work in progress, future work...
Methodology:
Automatic orientation: piecewise linear pushbroom model
GCP-based orientation: use Bayesian inference
Visualization: probabilistic contour maps
Fuzzy orthorectification
3D surface recovery from n images (n>2)
Uncertainty validation (sparse GCP, LIDAR points, accurate ref. DEM)
Applications:
Aerial coastal imaging:3D change monitoring(Intergraph DMC)
Data fusion from largesatellite data sets(Terra/ASTER)